How is the US economy really doing under President Trump? President Donald Trump has been boasting about the performance of the US economy ahead of the midterm congressional elections, claiming he has boosted incomes, reduced inflation and created jobs. Trump has repeatedly said: We have the hottest country anywhere in the world. While the overall US economy is faring better than many expected after Trump launched his sweeping Liberation Day tariffs in 2025, there's evidence those import taxes and the impact of the US-Israel war with Iran have pushed up inflation, with the harmful effects felt by most American households. The US economy is also being helped by an AI investment boom, and could falter if that spending were to dry up. So how does all this feel for the average American?

United States President Donald Trump has awarded the country's most prestigious honours in science and engineering to executives from top technology firms including SpaceX, Alphabet, Microsoft and Nvidia. On Thursday, Trump presented the National Medal of Science to billionaire SpaceX and Tesla CEO Elon Musk, Google co-founder Sergey Brin, Nvidia CEO Jensen Huang and Lisa Su of the semiconductor firm AMD. At the award ceremony, Trump credited the honourees with "helping pioneer an incredible future for America". "Because of these people here, we will always be on top. No one else can compete with us," Trump said.

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Denoising diffusion models have evolved into a state-of-the-art method for tasks in various fields, such as denoising and generation of images, text generation, or generation of synthetic data for training of other machine learning models. First hitting diffusion models (FHDM) are a particular class of denoising diffusion models with \textit{random} adaptive generation time tailored to generate data on a known manifold. Building on the conditioning framework of Doob's $h$-transform these models leverage the given information on the target data manifold to demonstrate strong performance across tasks while offering distinct features such as time-homogeneous dynamics of the generating process and a reduced average simulation time. Even though the theoretical investigation of standard forward-backward diffusion models has attracted much attention in the recent past, the statistical convergence properties of FHDMs are not yet understood. In this work, we show that, up to logarithmic factors, FHDMs achieve the minimax optimal convergence rate in total variation for spherically supported Sobolev smooth data distributions. In particular, this is the first statistical optimality result for denoising diffusion modelling with random generation time.

We can see that the error for the first term is mainly due to the sample approximation. We therefore refer to the first term as the Variance. We refer to the second term as the Bias. Our proof of convergence of the bias adapts the proof in [31, Theorem 6] and [11], and utilizes the fact that CY|X is Hilbert-Schmidt to obtain a sharp rate. A.1 Bounding the Bias In this section, we establish the bound on the bias.